Indices and surds

Find the exact value of

1. 32³⁵
2. (1681)⁻³⁴

Write √75 + √48 − √12 in the form k√3, where k is an integer.

Simplify fully (9x⁴)¹² × 2x⁻³

Write 123 − √5 in the form a + b√5, where a and b are integers.

Write 5 + 2√32 + √3 in the form p + q√3, where p and q are integers.

You must show all stages of your working.

1. Write 8^{x + 1} in the form 2^{ax + b}, where a and b are integers.
2. Hence solve the equation 8^{x + 1} = 4^{2x − 3}

Write 8x³ − 3√x2x in the form ax^p + bx^q, where a, b, p and q are constants.

Given that y = 3^x,

1. write 9^x in terms of y,
2. write 3^{x + 1} in terms of y.
3. Hence solve the equation 9^x − 4 × 3^{x + 1} + 27 = 0

Solve the equation

√5 x = √3 (x + 2)

giving your answer in the form a + √b, where a and b are integers.

Given that

2^x × 4^y = 18    and    27^x9^y = 2187

find the value of x and the value of y.

A rectangular tile has area 14 cm². Its width is (4 − √2) cm.

1. Find the length of the tile, giving your answer in the form (a + b√2) cm, where a and b are integers.
2. Show that the diagonal of the tile is exactly 6 cm.

1. Show that, for any positive integer n,

   1√n + √(n + 1) = √(n + 1) − √n

2. Hence find the exact value of

   11 + √2 + 1√2 + √3 + 1√3 + √4 + ... + 1√24 + √25